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permutation matrix

(Definition)

Permutation Matrix

Let be a positive integer. A permutation matrix is any $n\times n$ matrix which can be created by rearranging the rows and/or columns of the $n\times n$ identity matrix. More formally, given a permutation $\pi$ from the symmetric group , one can define an $n\times n$ permutation matrix $P_{\pi}$ by $P_{\pi}=(\delta_{i\, \pi(j)})$ , where $\delta$ denotes the Kronecker delta symbol.

Premultiplying an $n\times n$ matrix by an $n\times n$ permutation matrix results in a rearrangement of the rows of . For example, if the matrix is obtained by swapping rows and of the $n \times n$ identity matrix, then rows and of will be swapped in the product .

Postmultiplying an $n\times n$ matrix by an $n\times n$ permutation matrix results in a rearrangement of the columns of . For example, if the matrix is obtained by swapping rows and of the $n \times n$ identity matrix, then columns and of will be swapped in the product .

Properties

Permutation matrices have the following properties:

They are orthogonal.
They are invertible.
For a fixed positive integer , the $n \times n$ permutation matrices form a group under matrix multiplication.
Since they have a single 1 in each row and each column, they are doubly stochastic.
They are the extreme points of the convex set of doubly stochastic matrices.

"permutation matrix" is owned by Wkbj79. [ full author list (2) | owner history (1) ]

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See Also: monomial matrix

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Attachments:: example of permutation matrix (Example) by Wkbj79

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Cross-references: convex set, extreme points, doubly stochastic, matrix multiplication, group, invertible, properties, product, Kronecker delta, symmetric group, permutation, identity matrix, matrix, integer, positive
There are 18 references to this entry.

This is version 16 of permutation matrix, born on 2002-01-04, modified 2007-10-05.
Object id is 1232, canonical name is PermutationMatrix.
Accessed 12400 times total.

Classification:

AMS MSC:

15A36 (Linear and multilinear algebra; matrix theory :: Matrices of integers)

Pending Errata and Addenda

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